CC BY
85Electronic Journal «Technical Acoustics» http://webcenter.ru/~eeaa/ejta/
2003, 21
Abdelhafid Kaddour1, J. M. Rouvaen2, M. F. Belbachir1
1Universite des Sciences et de la Technologie d’Oran, Faculte de Genie Electrique,
Departement d’Electronique, B. P. 1505 OranM’Naouar, Oran, Algerie
2Institut d’Electronique et de Microelectronique du Nord, Departement Opto-Acousto-
Electronique, Universite de Valenciennes et du Hainaut Cambresis
Le Mont Houy, B.P. 311, 59313 Valenciennes Cedex, France
e-mail: kaddour_a@yahoo.com
Simulation and visualization of loudspeaker's sound fields
Received 04.12.2003, published 28.12.2003
The visualization of sound fields is useful for understanding the directional features of radiation of sound by an electro-acoustic transducer. The aim of this work is to present a numerical technique named here the Convolution Method (CM) for the simulation of sound pressure field radiated by a loudspeaker which can be represented by the simple concept of vibrating rigid piston located in a baffle. We developed a computational and visualization program for sound fields emitted by such transducer, which can be either driven by continuous wave (CW) or pulsed wave (PW). The numeric results of simulation for normalized contour surface map of acoustic pressure field are shown. For illustrating simulation results, the transverse acoustical beam section of sound pressure field produced by an electrodynamical loudspeaker is measured using intensimetry techniques.
Keywords: loudspeaker, sound field, convolution method, simulation, measurements, visualization.
1. INTRODUCTION
The knowledge of the directional features of the sound fields radiated by a loudspeaker is an essential characteristic of any audio system. The numerical calculation of the acoustic field radiated from a loudspeaker is a computer-aided tool in loudspeaker design and development. The resolution of the Hemholtz wave equation for velocity potential in a space filled with a fluid medium is the starting point in the calculation of the sound radiated from a surface. Thus the radiated sound field expression depends on the velocity distribution on the loudspeaker vibrating diaphragm. This velocity distribution is determined from a complex computation and vibration analysis. Several authors have expanded the velocity potential in spherical Bessel and Legendre functions, or used alternative methods to calculate the sound radiation from a loudspeaker [1-8].
The acoustic radiation from a rigid circular piston vibrating in an infinite rigid wall is a good model for studying a loudspeaker. The assumption of a piston-like motion is valid only at low frequencies. At higher frequencies, this approximation may not hold, owing to the breakup modes of the diaphragm. However, this assumption is still beneficial for
determination of loudspeaker’s sound radiation. Generally the starting point to compute the sound field at any spatial point inside a propagation medium is the evaluation of Rayleigh's surface integral [9], which is a statement of Huygens’ principle that the total sound field at an observing field point P into the propagation medium is found by summing the radiated hemispherical waves from all parts of the vibrating surface. It is difficult to solve analytically Rayleigh surface integral except in very special cases such as the acoustic pressure on the axis of a transducer. Various authors have proposed different theoretical approaches to transform this integral into an analytically tractable expression [10, 11].
The aim of this work is to present theoretical, computational and visualization solutions for the sound pressure field in air, produced by a loudspeaker, using a numerical method called here Convolution Method (CM). This method uses the convolution product of the acceleration function of the radiating surface and the impulse response for a specific field, which is deduced from the Rayleigh surface integral [12, 13]. We developed a computational visualization program for sound fields emitted by circular transducers. The sound pressure field can be computed for a surface vibrating either in continuous mode or impulse mode. In this work, numerical results of normalized contours of acoustic pressure field are shown. The transverse sound pressure beam of an electrodynamical loudspeaker is measured using intensimetry techniques.
2. CONVOLUTION METHOD: BASIC THEORY
The calculation of sound radiation pressure field emitted in a fluid by an acoustic source requires resolution of Rayleigh’s surface integral which is expressed as:
of vibrating surface S .
It is difficult to solve analytically except in very special cases such as the sound pressure on the axis of a loudspeaker. However Eq. (1) can be written as follows:
(1)
where O(r, t) is the spatial time-dependent velocity potential, c is the sound speed in fluid, r denotes the position of an observing point in fluid, t is time and v() is the uniform velocity
(2)
where * indicates the convolution in time-domain and <5(.) is the Dirac function. Thus the velocity potential can be expressed again as follows:
O(r, t ) = v(t )*O i (r, t),
(3)
with O t (r, t) is the spatial impulse response of the sound field for the transducer defined as:
(4)
3. SPATIAL IMPULSE RESPONSE FORMULATION
We consider a loudspeaker as a planar piston mounted in an infinite rigid baffle radiating into a homogeneous half space as shown in Fig. 1. Different methods based on geometric considerations exist for calculating the impulse response. According to Ohtsuki, the velocity potential impulse response is:
O, (t ) = -c- 0(t), 2n
(5)
where 0(t )/2n, which is named Ring function, represents the total angle 0(t ) = 2a(t) of an arc, including all points of the transducer equidistant to the observing point P, as shown in Fig. 1. The angle a(t) is determined by applying the law of cosine for the triangle ABC :
cos
(ct )2 + x2 - a2 - z2
2 x-\J(ct )2 - z2
(6)
Fig. 1. Equidistant points arc on the transducer used in the calculation of acoustic field
4. SOUND PRESSURE FIELD EXPRESSION
The sound pressure p(r, t) is obtained from the velocity potential O(r, t) as follows:
p(r, t )=PdO(zt^, d t
(7)
where p is the density of fluid. Substituting Eq. (3) into Eq. (7) the sound pressure can be expressed as:
P(r,t^37i(r,t). d t
(8)
From equations (5) and (8) the sound pressure can be expressed as:
p(r, t )= [j{t )*a{t)],
n
(9)
where —c is the impedance characteristic of the medium and y(t) is the acceleration function
of the radiating surface. In a previous work [14], different expressions of the acoustic pressure according to the position of point P have been calculated. It may be noticed that the acoustic field exhibits revolution symmetry around the z axis.
5. COMPUTATIONAL RESULTS
In this section we present the results of numerical investigation of acoustic pressure fields generated by a simple circular piston-like loudspeaker with a diameter D equal to 24 cm. The sound speed in air is taken equal to 342 m/s. The sound pressure field was computed in continuous mode (time-domain harmonic). The acoustic pressure calculated is normalized to its maximal value. As shown in Fig. 2.a and Fig. 2.b it is clear that at low frequencies, that is, ka < 1 (k = m/c = 2n/X is the wave number, X being the wavelength and a = D/2) the loudspeaker radiates the sound uniformly in all directions and behaves like a point source. The sound pressure falls as axial direction z increases.
a)
b)
Fig. 2. Acoustic pressure contours surface map: a) ka=0.1; b) ka=0.5
While when ka > 1 (medium frequency) the loudspeaker becomes directional and a little energy is radiated at other directions. On the other hand at higher frequency, that is ka=10, the sound field emitted by the loudspeaker becomes quite directive and secondary side lobes appear (see Fig. 3.b). Close to the transducer the interference effects produced in the nearfield govern the shape of sound beam.
a)
b)
0 20 40 60
Axial distance z (cm)
distance
Fig. 3. Acoustic pressure contours surface map: a) ka=5 ; b) ka=10
6. EXPERIMENTAL RESULTS
The visualization of transverse acoustical sound pressure distribution produced by an electrodynamical loudspeaker is performed using the acoustic intensity measurement technique, according to the standard ISO 9614-2. This technique consists in measure the active acoustic intensity at different points on a parallel surface at certain distance z from the front of loudspeaker vibrating surface, following a definite sweeping procedure. Taking account of practical criteria, a measurement grid of shape square 65x65 cm (x x y) including 169 surface elements forming a space matrix of 13 x 13 measurement points was chosen. The white noise emitted from the loudspeaker is captured by a sound intensity probe (Bruel & Kj^r type 3584) and is analyzed in 1/24 octave bands using a frequency analyzer (Bruel & Kj^r type 2144). The experimental result given in Fig. 4 is an example of measurement using the sweep technique.
a)
b)
Distance
Fig. 4. Acoustic pressure contours surface map for ka=1.1 a - Sagittal profile; b - Transverse profile measured at z=10 cm
We notice that measurement and simulation results are quite similar. The loudspeaker truly radiates the sound uniformly in all directions and behaves like a point source. The sound pressure is maximal on the axis of the loudspeaker and falls when one moves away of this axis.
7. CONCLUSION
A convolution method for computing the sound pressure field radiated by a loudspeaker driven by a continuous excitation is presented. The simulated results obtained using this convolution method are similar and agree with those obtained by using other methods [15, 16]. As an example, the visualization of sound fields of a loudspeaker computed using the convolution method and measured using intensimetry techniques are shown. We notice that results of measurements and simulations are very similar. This shows the effectiveness and validity of the convolution method. Finally, the method may also be applied to pulsed excitation, a study which is now underway.
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